All language subtitles for BBC.Magic.Numbers.Mysterious.World.of.Maths.3of3 - מוזר יותר ויותר - על מדע ומתמיתיקה

There is a mystery at
the heart of our universe.

A puzzle that, so far, no-one
has been able to solve.

I can't... This is too weird!

Welcome to my world!

If we can solve this mystery,
it will have profound consequences

for all of us.

That mystery is why mathematical
rules and patterns seem

to infiltrate pretty much everything
in the world around us.

Many people have,
in fact, described maths as

the underlying
language of the universe.

But how did it get there?

Even after thousands of years,
this question causes controversy.

We still can't agree on what maths
actually is or where it comes from.

Is it something that's
invented, like a language?

Or is it something
that we've merely discovered?

I think discovered.

Invented. It's both.

I have no idea!

Oh, my God!

Why does any of this matter?

Well, maths underpins
just about everything

in our modern world,
from computers and mobile phones,

to our understanding of human
biology and our place

in the universe.

My name is Hannah Fry
and I'm a mathematician.

In this series, I will explore
how the greatest thinkers in history

have tried to explain the origins
of maths' extraordinary power.

SHE LAUGHS

You've ruined his equation!

I'm going to look at how,
in ancient times, our ancestors

thought maths was
a gift from the gods.

How, in the 17th and 18th centuries,
we invented new mathematical systems

and used them to create

the scientific and industrial
revolutions.

And I'll reveal how,
in the 20th and 21st centuries,

radical new theories are forcing us
to question, once again, everything

we thought we knew about
maths and the universe.

The unexpected should be expected,
because why would reality

down there bear any resemblance
to reality up here?

In this episode, I explore paradoxes
within modern mathematics.

Who shaves the barber?

And I discover the very weird
worlds that maths seems

to be leading us into.

SHE SCREAMS

Maths is very much part
of our modern world.

Even the images you're watching now
are essentially numbers

processed by computers.

Sorry, guys.

Would you mind taking a photo of me?

Oh, sure. Give me one second.

Today, maths is at the heart of big
business, in the development

of new software, such as facial
recognition technology.

All of which, fundamentally,
is based on mathematical algorithms.

And it matters because copyright
issues and legal ownership

can depend on where
that maths comes from.

You can phrase
the question like this -

is maths a genuine, fundamental
part of our universe,

something that we have discovered?

Or is it merely invented?
A language that we've created

just to describe
the world around us?

THEY SCREAM

Mathematicians have argued
over this idea for centuries.

And even today, this question
is a thought-provoking

and challenging dilemma.

So far, I've explored how,
in ancient times, maths was revered

Perfect, complete and gratefully
discovered by humans.

But through the ages,
new areas of mathematics,

like algebra and the concept
of zero,

have, quite simply, been invented.

But for most of us, we normally
think of maths as just a series

of objective facts based
in logic that someone,

somewhere has discovered.

Facts that we all start
to learn at school.

If you're anything like me,
you'll remember maths at school

being taught as a series of rules.

It was very logical, it was very
ordered, very complete.

Very black and white.

There were right and wrong answers,
which you didn't necessarily get in

other subjects like art or
like music,

which were much more about
preferences, about opinions

and about cultural differences.

It felt like the mathematical rules
were intrinsically true.

But why?

What are the fundamental
mathematical laws?

To answer that question,
you have to categorise everything.

You have to boil maths
down into distinct groups of objects

in something called set theory.

Set theory is a language
that talks about groups,

or sets, of items.

So, for example, the set of odd
numbers are all the whole numbers

that cannot be neatly
divided by two.

And the set of even numbers
are those that can.

This reveals a basic rule.

Adding an odd number to an even
one produces an odd number.

From simple rules like these, you
can build up more and more

complex rules and relations of
maths.

But there's a problem
with set theory.

A paradox at the heart
of mathematical rules which caused

a bit of a crisis at the start
of the 20th century.

You can discover this paradox
yourself by going to your local

hairdresser or gentleman's barber,
and trying to define what you find

in a concise and complete way.

Hello. Hello.

I was wondering
if you could help me.

I am looking for the very
definition of a barber.

I think I can help with that.

Mathematicians took the same
approach to precisely define

the laws of maths.

So, if you were looking
it up in a dictionary,

that one sentence that defined
what a barber was,

what would you say it was?

Cut men's hair.

But that could be a hairdresser
though, right? A hairdresser?

It needs to be a unique
definition for barbers.

Barbers, and only barbers.

Cos there's the shaving element,
as well, isn't there?

Yeah, that's true. I've never
had a shave in a hairdresser's.

No. That's true. I've had a chat.
The chat?!

SHE LAUGHS

Yeah. It's a fair point.
It's a very important part of it.

You do hear some stories,
being a barber.

So, actually, I suppose,

the shave thing is something that
only barbers do. Mm-hm.

So, someone who shaves men.

But a barber doesn't shave all men.

And I need a phrase that uniquely
and completely identifies a barber

OK, let's see where we are, then.

So, we've got, a barber shaves all
men, but only the men who shave

but don't shave themselves?

Yes. Yes.

All right, I think we've settled
on something now.

We've agreed on...

Sound about right?

I mean, it doesn't exactly
roll off the tongue!

But I think it's fairly accurate.

But, hang on a second!

There's a bit of a paradox here.

Who shaves the barber?

Well, can a barber not shave
himself?

But if he does shave himself,
then our

definition here says
that he doesn't shave himself.

Let me clarify that.

If he doesn't shave himself,
then according to the definition,

he's one of the men
shaved by the barber.

So, he does shave himself.

Attempting to create
a mathematically precise definition

creates a contradiction
where the barber both shaves

himself and doesn't shave himself.

First, the bristles...

This is known as
the barber's paradox.

You've got it!

I want to do it perfectly!
Perfect, OK.

It is an illustration of the paradox
at the heart of mathematics,

which was discovered in 1901 by one
of my favourite troublemakers,

Bertrand Russell.

The problem for maths
was that Russell's paradox

undermines the logic
of defining things,

like odd or even numbers,
by putting them into

categories or sets.

Over here, I have got a set of
clipper attachments.

And in there, I have got a set of
things

that aren't clipper
attachments.

Clipper attachment goes in there...

Not a clipper attachment,
goes in there.

Clipper.

Not a clipper.

Now, the question is -
where does this bag belong?

It's clearly not
a clipper attachment.

Is it going to attach to a clipper?

No, it's not, which means
it needs to go in there,

but we've got a problem,
because this sink is supposed

to only contain things that are not
clipper attachments.

Which means that the contents
of the bag can't go in the sink.

Since the bag, or set, is not, in
itself, a clipper attachment,

but, by its definition,
contains clipper attachments,

we can't easily categorise
where the set belongs.

Similarly, the barber can't,
in a logically consistent way

be contained in a set
of people that do

shave themselves or the set
of people who don't.

Russell's paradox shows
that there is a logical problem

with trying to categorise
anything into coherent sets,

whether it's barbers,
clipper attachments,

or even numbers, and this logical
puzzle exposed a fault in

the bedrock on which all the rest
of maths is built.

If the foundations are shaky,
how can we trust everything else?

Bertrand Russell realised
that mathematics was on much shakier

ground than people had originally
thought.

It turned out to be much, much
harder to really lay

a solid foundation for maths
that everybody agreed on,

and this is still wonderfully
controversial to this day.

That's what you do
in science and mathematics.

You take a sledgehammer.

You smash at whatever structure,
whatever edifice you've built.

You try to find the weaknesses
and that allows you to figure out

what needs to be shored up.

And that's really, I think,
the legacy that Russell left us.

I think of it as...in some ways,
the death knell,

or at least a major challenge,

the attempt to ground
mathematics in logic.

And that's the thing that
becomes really hard

in light of Russell's paradox.

Russell's paradox caused a real
crisis amongst mathematicians.

Suddenly, maths was uncertain.
It was fallible.

And if it has these
fundamental problems,

how can it possibly be discovered?

So, does that mean that maths
has to be invented?

Just a human language

and all of the flaws
that come with it?

If maths is merely an invention
of the human mind,

it's perhaps not that surprising
that it's not perfect.

But I don't think that I'm ready
to accept the invention

argument quite yet.

Maths just seems to be too good
at predicting the behaviour

of the world in ways that we never
could have imagined.

Because, just as Bertrand Russell
was exposing

the limitations of maths in one way,
another titan of the 20th century,

Albert Einstein, was pulling it back
in a completely different direction.

BABY GIGGLES

Take what is probably the most
famous equation in the world.

With just five symbols,
it looks so simple.

It's almost childish.

Yet, it contains some incredibly
powerful mathematical

and philosophical concepts.

I'm talking, of course,
about E = MC squared.

So, "E"...that's energy.

That is equal to...

.."M", that's mass.

Times by a constant, C.

It's the speed of light...

..squared.

There is so much more
to this equation than meets the eye.

It is Einstein's discovery
that matter and energy

are equivalent, and that
has profound consequences.

This equation gives us one
of the immutable laws

in the universe -
that nothing can travel faster

than the speed of light.

Try this one.

The reasoning is this...

Making something move requires
more energy than keeping it at rest.

And because this "C"
here is a constant,

if the energy goes up by
accelerating something,

the mass also has to increase.

So, that means that you or I
actually weigh a tiny bit more

when we're
moving in a car or a plane.

The increase in mass only becomes
significant when objects are moving

at speeds close
to the speed of light.

As an object approaches
the speed of light,

its mass rises faster and faster,
which means it takes more energy...

..to accelerate it further.

It can't, therefore, reach the speed
of light, because the mass becomes

infinite, and it would require
an infinite amount of energy

to get there.

You've ruined his equation!

As well as proving there's
a cosmological speed limit,

this single equation also explains
how all the stars in the universe

convert mass into energy
as they burn brightly

in the night sky.

Einstein's famous equation
has proved itself to be a remarkable

match for reality every time it's
been put to the test.

Einstein had uncovered one
of the essential mathematical rules

underlying the cosmos.

It seems like clear
evidence that that maths,

at least, is discovered.

But Einstein didn't stop there.

Using the power of mathematics,
he brought about a fundamental shift

travels through space.

To see that evidence for myself,
I've come to an observatory to do

some serious thinking
about what we actually see

when we look at stars
in the sky, such as our sun.

If things were happening right now,
we wouldn't be able to see

it until 8.5 minutes
later, because that's how long

it takes the light
to travel to the Earth.

So, when you're looking at the sun,
you're seeing how it was

8.5 minutes ago? Exactly.

And objects that are further away,
we see them as they were further

back in time.

So, for instance, there are other
stars in our galaxy

that are thousands
of light years away,

so we see them as they were
thousands of years ago.

So, when you look in a telescope,
and you're seeing them

how they were when people
were building pyramids

and Pythagoras was discovering
his rules on Earth.

Exactly. And we can even see
things that are

even further away than that.

So, galaxies outside our own galaxy.

We see many of them as they were

a billion years ago or more.
Gosh. Goodness.

Does this work at smaller scales,
then? Is there, like, a limit

to how big something has to be
before this works?

If you...

I mean, I'm looking
at you now, right?

Light, presumably,
is taking time to bounce

off you and for me to see you.

Yes, it is.

But light travels at
an incredibly fast speed,

300,000 km per second, roughly.

So, the time it takes to travel
from me to you is very, very tiny

fraction of a second.

But, in theory, I am
seeing you in the past.

In theory, yes, you're absolutely
seeing me in the past.

All of this shows that we can never
know what the universe

is like at this very instant.

The universe is, remarkably, not
a thing that extends just in space,

but in time, as well.

This is fundamental to Einstein's
revolutionary insights

about our universe.

He realised that the very concept
of time is relative.

That is to say, it depends on

the position and movement
of the observer.

He worked it out by thinking
about events that appear

to be simultaneous.

So, let's imagine that you're
in a hot-air balloon floating

above the observatory here,
and you're high enough

that you can see a flash of light
in London, say, and another

one in Portsmouth.

And let's assume that these flashes
of light go off such that you see

So, from where I am, it looks like
they are both flashing

their lights at the same time?

At exactly the same time.

But if I were in an aircraft
that was flying very fast

towards London, I would see
the flash of light in London

before the flash of light
from Portsmouth.

Using the inescapable
logic of mathematics,

Einstein realised that if
an observer is moving towards one

of the flashes, they would see
that flash before the other one

caught up with them.

So, for them, the flashes
are not simultaneous.

But who's...? OK... I mean,
they did go off together.

Who's right?

Am I right in the hot-air balloon?

In fact, there is no way
of saying that you are right

and I am wrong in how
we observe these events.

This is called relativity.

So, our whole concept of time,
our whole concept of time

means what happens first,
what happens second,

comes down to where we are
and how we're moving.

Exactly.

So, the concept of time is now
inextricably linked to the positions

in space and your movement
through space.

So, this is why we can't describe
space and time separately,

but we have to put them
together in space-time.

You can't separate the two.

And that all comes down to this idea
that Einstein managed to prove

via thought experiments.

Yeah, that's the amazing
thing about it.

Purely through thought
experiments and...

And a good bit of maths.
And a good bit of maths.

A very good bit of maths, yes.

Einstein was using the mathematics
to make sense of the universe,

and claiming that the universe
was nothing like what anyone

thought it was.

His concept of relativity flew
in the face of what people

had believed about space
and about time for centuries.

Whether that was the Greeks thinking
that the universe was eternal

and unchanging,

or Isaac Newton's more mobile and
mechanistic descriptions.

Einstein took his thoughts even
further, attempting to wrestle

gravity into a neat
mathematical law.

He believed it was all
down to the strange

behaviour of space-time,

and if he was right, as he laid out
in the theory of general relativity

in 1916, then gravity
will even affect light.

If you've got a star shining
light from over here,

then you, the observer,
over there, will receive

it in a straight line.

But, if there's a massive
object in the way,

you might think that you won't be
able to see the star.

However, Einstein predicted
that the mass of an object

will distort the space-time
around it, and anything moving

through that warped space-time

will have to follow the curves.

This warping of space-time,
Einstein said, is what we usually

describe as gravity.

We think of gravity as keeping
the planets in orbit around our sun.

In fact, he said, it's the result
of the distortion of space-time

near massive objects.

And Einstein calculated the precise
effect it would have on light.

So, the starlight, while still
technically travelling in a straight

line, will follow the curves of
space and appear around the object.

Einstein predicted that,
in exactly this way,

we should be able to observe light
from distant stars getting bent

as the stars pass behind our sun.

But a theory is just a theory,
an invention of the mind.

It only becomes a discovery
when proven by practical

measurement or experiment.

In the decade after
Einstein's prediction,

solar eclipses around the globe gave
scientists the chance to repeatedly

test his theory.

The darkness of the eclipse allowed
them to actually see stars passing

When scientists took the
measurements,

they discovered that light from a
distant star

was bending around the sun in
exactly the way

that Einstein had predicted.

The mathematics of general
relativity was correct.

With general relativity,
Einstein completely upended our

understanding of space,
time, matter, energy,

and kind of what else
is there to the nature of reality.

All of a sudden, we learn
that mass and energy can warp

the fabric of space and time

in this beautiful,
interconnected dance

where the motion of matter affects
the warping of space and time,

which affects the motion
of other matter.

We used to think of
space as this boring

static stage upon
which events unfolded.

Then Einstein told us that space is
itself an active player

in this game, like
a stretchy rubber sheet.

And, yet, a substance
perfectly described

by beautiful mathematical equations.

I mean, how did he think of that?

How did he think of
something like this?

Einstein's description of gravity,
the warping of space-time,

accurately explains why
objects stay in orbit,

whether they're satellites around
the Earth or galaxies

around black holes.

His equations are being tested
and reproven every day,

and without Einstein's general
theory of relativity,

modern communication, GPS or
satellite TV systems

couldn't even function.

Although this theory
came from his mind,

from thinking about the problem,
rather than from real-world

experiments, it's still
so good at predicting,

so perfectly capable of describing
what happens in the universe,

that it must be reflecting some
underlining mathematical truth.

And this lends quite a lot of weight
to the argument that mathematics

is discovered, which is something
that matches up with my own

experience, because when you're
toying around with mathematics,

it really does feel
as though you're exploring something

that already exists.

But if we accept that maths does
already exist and is an intrinsic

part of nature, then surely
all the rules are out there waiting

to be discovered.

In some ways, mathematics
is quite a lot like

a game of chess.

So, you have these very strict rules
that you're not allowed to break,

but within those rules,
there are all kinds of opportunities

to play around and be creative.

The only problem is that, in maths,

no-one tells you
what those rules are.

We have to work them
out for ourselves.

Most mathematicians
like a challenge,

but this idea got blown apart
at a maths conference in 1930

in the Prussian city of Koenigsberg,
when two great mathematicians

and their conclusions collided.

On the one side,
you have got David Hilbert,

a mathematical king in every
possible sense of the word.

This is an enormously
well-respected man who laid down
the gauntlet,

asking people to come up
with a fundamental set of rules

on which every mathematical
proof could be based.

On the other side was a young
academic called Kurt Godel.

In contrast to Hilbert,
who thought that mathematics

should be built from
the ground up by humans,

Godel thought that
mathematics was discovered.

He believed that mathematical
truths exist outside of us,

and that we have very little say
in what we can find.

That summit in Koenigsberg
can be seen as a clash

between those who thought that
mathematics is part of our fabric

of reality to be discovered,
and those who saw it as a language

under our control.

Hilbert was confident that humanity
would soon know all there is to know

in maths.

But Godel, who had also been trying
to find

the rules of maths,
had come to the opposite conclusion.

In a side room at the summit,
Godel quietly announces that,

in fact, however hard you try, there
are always going to be some

things that are unknowable.

There are always going to be parts
of the mathematical game

that can't be fully explained.

And if you can't know all the rules,
how can you play the game?

According to Godel, any rule-based
maths system is always

going to have some things
that are either unknowable

or unprovable, and what's more,
he could prove it -

which is kind of ironic, if you
think about it.

This was quickly accepted,
and became known as

Godel's Incompleteness Theorem.

And it puts an interesting twist
on our key question.

It shows that, even if mathematical
rules truly are part of the universe

and we're simply discovering them,
we are nevertheless going

to have to accept some
of those rules without knowing

how or why they are true.

Normally, people think
that there's some intrinsic

difference between science and math
on one hand, and faith-based belief

systems on the other,
and what Godel's theorem

tells us is that's not true...

That there are things in mathematics
that you have to take on faith

or you can't do the mathematics.

To me, this was an astounding
thing to realise.

We're going to have to accept
that we can't give maths

a foundation in formal laws
or in logic

in the way that we thought we could.

I think it's enormously exciting

that math, in some sense,
is open-ended.

So, in a sense, it puts an end
to one way of thinking

about mathematics, but I think, it
actually adds colour and richness

to the subject because it's just
going to keep on going.

So, what does Godel's Incompleteness
Theorem mean for our view

of the universe and the parts
that maths plays in it?

Well, it depends on what
you're trying to use maths for.

If your goal is to use
it to describe what's around you,

then it still offers
a very detailed picture -

enough to navigate your way
through the universe

and to explain its features.

Sure, the map is not going
to be the same as the terrain,

but even if maths is a bit
incomplete around the edges,

you could argue that it
doesn't really matter.

Although Godel proves it's not
possible to formalise all of maths,

it is possible to formalise all the
mathematics we actually need to use.

Take flying as an example.

Now, I did my PhD in the mathematics
of aerodynamics,

and that means I spent four years
poring over equations

for wing sections and wind speeds.

It's stuff that I know
like the back of my hand.

But does that qualify me for going
up in one of these on my own?

Absolutely not!

And on the other hand,
these guys don't really need to know

any of this stuff to make them
graceful acrobats in the air.

Not having a complete understanding
doesn't always matter.

We've still flown successfully
for over 100 years.

And now, it's my turn.

And then this is your...
Diagonal line.

The strap that comes across.

This will dig in a little bit
on take-off when you're leaning

forwards and running down the hill.

I can handle it. It should be
a little bit uncomfortable.

I can handle it. Don't worry
too much about it.

And do you have quite a good feel
for where the thermals are?

You have to have the
right weather conditions.

So, if you imagine a hill that faces
totally into the wind,

that's well drained, maybe darker,
and it will create this

kind of pool of warm air, and then
it will... Once it kind of reaches

a decent temperature difference, it
bubbles up through the atmosphere.

Yeah, it's almost like we've got
kind of opposing skills. Yeah.

And, like...they're sort of about
the same thing, but they...

You don't need my skills
to do what you do,

and I couldn't do what you do.
Mm-hm.

I guess the ground-speed element
has a bit of maths in there. Yeah.

I always thought the lesson bit of
maths to begin with.

Where's the wind coming from?
How strong it is.

How fast am I going to go,
if I'm pointing into the wind?

But you're not solving
Navier-Stokes equations, are you?

I don't even know what that means!
Yeah, exactly.

THEY LAUGH

Before the theoretical analysis
of aviation came along,

the practical side of flying was
mere trial and error.

Now, we have a much
more reliable understanding

of what keeps us aloft, and it
doesn't really matter

if the maths behind it is,
ultimately,

a bit fuzzy around the edges.

In the real world, the best that we
can do is just accept

Godel's Incompleteness Theorem
and get on with life.

Hey!

It's amazing.

There's a thermal! Yeah!

Woo! Woohoo!

You're feeling much more alert.
A bit stronger? Yeah.

We have to put aside,
for the moment,

the question of whether maths
is invented or discovered,

because it now looks like

we may have to determine which part
of maths we're asking about.

You see, for me, Godel's work
highlights the distinction

between pure theoretical maths and
practical applied maths.

So, here is how I see things...

With mathematics, there's a split
down the middle of the subject,

because the story changes depending
on what world you start with,

whether it's the real one or one
that exists in our imaginations.

And right now, when we're flying,
this is very much in the realm

of applied mathematics, where
everything is tangible and practical

and a little bit imprecise.
But, alongside that,

is where the more theoretical pure
mathematics lives.

That's where you have your proofs,
your paradoxes,

and incompleteness theorems.

A realm which doesn't match
up with a physical reality.

A sort of imperfect perfection.

Even though I instinctively feel
that maths is discovered,

I like that there is this pure
theoretical part of maths

that isn't found in reality.

And, since the maths there
doesn't need to match reality,

it's a convenient place where we can

leave all the weird contradictory
bits that we come across.

Yeah! Woo!

However, I might have it
the wrong way round.

Although pure theoretical maths
seems rather divorced from reality,

that might merely reflect
the fact that reality is not

quite what we think it is.

And it's a reality that we can
uncover through the strange maths

of quantum physics.

The weirdest worlds that most of us
have come across are likely

to be in fiction, such as this,
Alice's Adventures In Wonderland.

The author, Lewis Carroll -
real name Charles Dodgson -

was actually
a mathematics don at Oxford,

and a staunch traditionalist.

It's generally believed that much
of this surreal story is a thinly

veiled satire on the new avant-garde
maths that was flourishing

when he was writing in the 1860s.

Still feels relevant today,
and applies equally well

to the new weird kid on the block,
quantum physics.

Take a close look at
the physical world around us,

and you can reduce it all to maths.

The solid bricks of our houses
or the blood cells in our veins

can all be reduced
down into chemicals,

which comprise elements,

which themselves are made
up of atoms, comprising a tiny

nucleus of protons and neutrons
and electrons buzzing

around in a cloud
of mostly emptiness.

The protons and neutrons in turn
are built from smaller subatomic

particles that we can't
directly observe.

We can only verify their existence
using experiments and mathematics.

As we delve deeper into this world,

scientists have discovered something
very strange indeed.

We can never actually know
the precise location

of most particles in this
subatomic or quantum realm.

All we can know is the likelihood
of them being somewhere,

a mathematical formula that
describes the probability

of their position.

All of this means we are,
fundamentally, at a quantum level,

just a great fuzz of
energy and probabilities.

I'm not sure Lewis Carroll
would have liked that.

And the only way to explore
this ill-defined quantum world...

Oh! Hey. Hello.

..is through mathematics -

perfectly equipped to handle
strange probabilities.

It seems like there's quite a lot
of uncertainty in quantum physics.

Does that bother you?

Um, no.

When I heard that things were, you
know, uncertain,

and also against our common sense
in quantum physics,

then I thought, like, "Oh, wow!
That sounds interesting.

"I want to know more about that."

The pivotal maths behind the quantum
world was first laid out by Austrian

physicist Erwin Schrodinger in 1926.

His equations accurately describe
the unusual behaviour

of subatomic particles.

OK, all right,
I'll tell you what, then...

Quantum physics lesson 101,
where do we start?

Give me the lesson.

Um, OK, I would say we would
have to start with superposition.

So, let's talk about electrons.

So, they're a very small particle,
and they can be in two states.

They have a state called the spin,

and the spin can be pointing
up or down.

So, if we were in the classical
world, the spin could only be either

up or down.

But in the quantum world,
the spin is in a superposition,

which means it can be up and down
at the same time.

Let me see if I understand this,
then.

So, superposition is where something
is and isn't something

at the same time? Yes.

We can think about some examples.

So, let's say that we have a cup,
and the cup is full of water

or... That's one state.

Another possible state
is that the cup is empty.

So, if we were to bring the quantum
ideas to the classical world,

we would say the state,
one possible state of the cup,

would be to be empty
and full at the same time.

OK, which you never see
in the world that we're living

in, you never see a cup
that's full and empty.

Yes, we don't.

But you see this a lot
in the quantum world?

Yes, superpositions are an essential
part of the quantum world.

Like a light being on
and off at the same time.

Exactly. Or the cake being eaten or
not eaten at the same time.

OK. It's a very tough idea to get
around. Yes, yes.

Given two possible outcomes,
in the quantum world,

we now have to allow for a third one
- the combination of both outcomes.

At the quantum scale, you can have
your cake and eat it.

This is such a weird idea.

How do we know it's real?

Well, because we've done many
experiments to prove it

that show exactly that behaviour.

What does that experiment look like?

Well, if we put it, say, in terms
of things we have here on the table.

We could think about... Let's say
that I wanted this piece of sugar

to come into my cup,
but there is this pot in the middle.

So then, if the sugar is going to
come from here to my cup,

it could either go this way or that
way in the classical world.

But in a quantum experiment, it can
take both routes at the same time,

and I would be able to distinguish
that it did that

if I did a quantum experiment.

I can't.... This is too weird!

Welcome to my world.

So, you go through this whole
transition, from first,

the ideas and the mathematics,
and then up to showing

it in the experiment.

What came out of Schrodinger's mouth
was a prediction of something

even stranger that can sometimes be
produced when particles interact

in the quantum world -
a phenomenon called entanglement.

All right, tell me
about entanglement, then.

OK, so take two electrons.

If the electrons are entangled,
and I do something to one

of the electrons, for example change
the direction of the spin,

that will instantaneously affect
the state of the other electron,

even if they are separated long
distances.

How far away are
they from each other?

Well, they can be
a few centimetres...

but now the latest experiments,

they're using satellites, show
entanglement across 1,200 km.

What? Yes.

You've got something over here,

and you do...? And something
1,200 km away.

You do something to one,
and it instantly...

The other one instantly
knows what's happened?

Yes, you'll affect the state
of the other one instantly.

Apparently, there is
no cause or link.

The only thing we can say is that
the two particles are synchronised.

How does one know
what the other one's doing?

Well, that we're still trying to
understand, because that's

what mathematics tells us,
and then we can show

it in the experiment,
but we're still are struggling

to understand what that means.

And one of the reasons
why we don't understand it...

And, you know, like you're asking,
is because we don't see

it in our everyday lives.

So, let's say it's not part of our
experience and common sense.

But that doesn't mean it doesn't
happen.

I don't want you to go crazy...

So, quantum mathematics
has made predictions,

which have been
discovered to be true.

But, despite that,
the quantum world is so weird,

it suggests to me that the maths
behind it is just invented.

It feels like what we're seeing
is evidence of a man-made system

being pushed too far.

These are the absurdities that
appear when it's applied

to situations it
wasn't designed for.

But my quest to find the truth
about maths takes me back to nature.

There is amazing new evidence
that quantum processes

might actually be crucial to our own
existence,

and much of life on Earth.

That would strengthen the argument
that mathematical processes

are intrinsic to our world.

That maths is discovered.

It all comes down
to the photosynthesis,

the process that converts sunlight
into chemical energy used in life.

It takes place in the molecules
called chlorophyll,

which can be found in plants,
algae and bacteria.

In bacteria, we have something
that's similar

to what we have in plants.

So, this is the stuff
that captures the sunlight?

Exactly. Each of these molecules,
each of these little blue things

here that I'm showing, is bacterial
chlorophyll, and if we take

it apart, it will capture light.

The chlorophyll captures light
by absorbing particles

of light, or photons.

So, a photon is absorbed,
and it's absorbed by all of them,

so energy shared by all of these
bacterial chlorophylls,

and that sharing, we call it
using a quantum superposition.

Because it's coming in and hitting
one of these,

but all of them are somehow...

In a way, it's as if each of the
electrons of the chlorophylls

are talking to each other
and sharing the energy around.

The subatomic particles
in the chlorophyll are synchronised

in a way that can only be described
by quantum mechanics.

Does it do a good job?

I mean, is it efficient?

That is part of why photosynthesis
is efficient.

Because, by sharing the energy
among all of them,

it's easier to transfer the energy
to another molecule.

Imagine if you have to share
the energy one by one,

you have to explore
each part separately,

but if you share the energy
all together, you explore

all the parts at the same time.

Every leaf on every
plant on the planet

has been following these quantum
rules for millions of years.

And we still don't fully
understand how they do it.

Without quantum physics,
despite all the mathematical

uncertainties and ambiguities,
plants wouldn't produce

oxygen so efficiently.

And without oxygen,
we wouldn't exist.

The systems are amazing,
because they are effectively

the interface between using a little
bit of classical mechanics

and a little bit of quantum
mechanics to operate

in a wonderful way.

Ultimately, quantum mechanics
is at the heart of photosynthesis

and, well, I guess,
all of life on Earth.

It is. It is.

We can say life is nothing but
quantum mechanics giving us energy.

So, what does all this mean
for our key question

about the origins of maths?

There is no shortage of evidence
that mathematical rules

are intrinsic to the world.

We keep discovering them everywhere.

However, we now know we have to take
some of the maths on faith,

and believing in the numbers is
taking us to a very strange world,

with crazy notions like
superposition and entanglement

at the core of it.

Quantum mathematics is inextricably
linked to the world as we know it.

Or, as we knew it.

Because the world is actually
a whole lot weirder than we thought.

What quantum mechanics does do is
force us to question what is real.

And what is reality, anyway?

Just how much light
can mathematics shed on reality?

With the world stripped bare,
exposing the nuts and bolts

of existence,
what does maths tell us

about this realm of subatomic
particles?

The maths that underlies it
isn't particularly pretty,

but it can all be written out in
just one equation.

This is the formula that describes
the constituents of the universe.

It has become well enough accepted

to be called the standard model
of subatomic physics.

I told you it wasn't pretty.

Now, you're just going to have to
take my word for it on this one.

This equation encapsulates
all of the fundamental properties

of the subatomic world.

But there are a couple of sticking
points.

For one thing, no-one has ever
satisfactorily explained

how our common-sense, day-to-day
version of the world emerges

from this kind of subatomic reality.

All of that fuzziness, all of that
uncertainty in the quantum world,

just how does it end up giving us
that comfortable,

familiar solidity of the normal
world?

At the other end of the spectrum,
the solar system and beyond

is beautifully and accurately
described by a different equation.

Einstein's general relativity.

This remarkable equation
tells you about gravity,

about the warping of space-time,
about general relativity.

And when you take these two
together,

these two single mathematical
sentences,

they are enough to tell you
everything you need

about the fundamental behaviour of
the universe and everything in it.

There is nothing more
articulate than mathematics.

Maths seems to be written
into the physical universe.

So, on the one hand,
at the teeny-tiny scale,

the standard model of particle
physics does this amazing job.

And in the ginormous scale,
general relativity,

I mean, you couldn't ask for
anything more.

There's just one problem when you
try and put these two together.

The problem is that general
relativity breaks down

in the quantum world.

Gravity simply doesn't apply
to particles at the subatomic scale.

Meanwhile, quantum effects
are virtually never seen

at the scale of humans and planets,
where gravity rules.

You and I are never
in a superposition of existing

and not existing at the same time.

So, what does this mean for us?

Are there two different worlds,
each obeying their own sets

of mathematical laws?

Solving this conundrum is one
of the biggest problems

that puzzles scientists today.

Will we ever reconcile the two?

I think it's perfectly plausible
that, within our lifetime, somebody,

the mathematical structure
which unifies Einstein's theory

of relativity with quantum mechanics
and just provides the perfect

description of this world.

And that would be really exciting.

Will we have one?

How do I know?

We would all like to have one.

But, you know, maybe we are not
smart enough

to formulate a theory that combines
everything.

It's hard.

I do believe that there are good
ideas out there and that eventually,

it might take a long time,
but, eventually,

humans will work this out.

I'm confident about that.

So, will we make it all the way
to include all possible forces

at all possible scales with all
possible forms of matter?

It's a hope I have for our species,
that's all I can say.

The incompatibility
of these two great theories,

general relativity
and quantum mechanics,

creates a serious obstacle
for believing that maths

is really discovered.

And there's a bigger hurdle to come.

Many of the best proposals to unify
general relativity and the quantum

world have consequences
that are even weirder

than the problems
they're trying to solve.

They predict the existence
of multiple universes.

This idea is rooted
in the mathematical explanations

of the quantum world, and the work
of its founding father,

Erwin Schrodinger.

The mathematics in Schrodinger's
equation insists that particles

can exist in multiple
states at the same time.

And Schrodinger himself says
that these possibilities aren't

just alternatives,
but really happen simultaneously.

This can lead to multiple universes.

And the maths also suggests
there's an infinite number of them,

each slightly different
from the others.

Thank you.

ALL: Mathematically speaking,
in an infinite universe,

everything that's
possible has to happen somewhere.

Yeah, that's right.

Everything possible happens.

Somewhere.

Even Schrodinger acknowledged
that the consequences

of his equation describing the
quantum world might seem lunatic.

But if there's one
thing I've learned,

it's that you should
trust the maths.

So, maybe our experience
isn't special,

maybe our reality isn't unique
after all.

There are so many distinct avenues
of investigation that lead

to the possibility of a multiverse.

From our studies of unification
and string theory,

from our studies of quantum
mechanics, even from the study

of space going on infinitely far.

Even that gives rise
to a version of the multiverse.

If we're going to reject everything
that just seems weird, we're almost

guaranteed to reject the true
theories of the future

when they get discovered.

I think we should just chill out,
accept that the world is weird

and that's just part of its charm.

And trust the math.

So, why does all of this matter?

Well, if maths
really is discovered,

then there is an intrinsic truth
behind the maths we uncover -

however weird that truth
seems to be.

If maths is invented,
then how do we know

what is true or false?

Is it true purely
because we define it so?

And how does it relate
to the real world

that we all experience?

Clearance...

In this series, we've seen that
maths can explain

so much of our world,

from aerodynamics to planetary
orbits, from the subatomic world

to processes crucial to life
on Earth.

And that is something I just can't
accept as a coincidence.

So, here's my take on things.

For me, it's almost
as though you have this alternate

parallel mathematical world
that hides just beneath our own.

You can't see it,
you can't touch it.

The only way that you can explore it
is by using the language

that we've invented.

All of those symbols and equations
and conventions are our only tools

of navigation, and they are
undoubtedly man-made.

But once you're inside that world,
once you're exploring the landscape

that mathematics has laid
out in front of you,

I am absolutely convinced that you
are on a voyage of discovery.

It is a world without a
human designer.

So, ultimately, I think it's both.

Mathematics is a little bit of
invention and a lot of discovery.

Mathematicians will probably never
all agree, and maybe we will never

find a definitive answer,
but the consequences

of having that debate
is why it really matters.

We have used mathematics for a much
deeper understanding of nature

and of the universe in general.

We know about the universe now,
things that,

a few hundred years ago,

people didn't even know what to ask.

Searching for the truth about maths
has, over 2,000 years of history,

transformed the human experience.

Discovering patterns everywhere
in nature has given us structure

and beauty and inspiration.

Inventing new areas of maths has led
to an explosion of technology

that, ultimately, underpins modern
trade and computing.

We have discovered powerful rules
that we continue to use to explore,

enhance and explain
the world around us.

And we have had a tantalising
glimpse of what could be to come.

It's quite possible
that what we have been doing

in science for all the centuries is,
in some sense,

looking for our keys
under the lamp post.

We have been able to use mathematics
to describe what happens out there,

but that could be the tip
of an iceberg of reality

that we as yet don't have any
understanding of,

haven't yet had any contact with.

But most of all, I think that asking
questions about the origins

and truth of maths has given us a
purpose,

it's given us understanding.

Ultimately, maths has given us
meaning.

What is it that makes our world work
the way that it does?

Explore more about the magic
and mystery of mathematics

and how it impacts our everyday
lives by going to the web address

below and following links
to the Open University.